Mach’s Principle and Frame Dragging
We will begin by quoting the objection to the Kerr metric raised in the published papers of Rouke and MacKay, which requires them to develop a new metric for the rotating black hole, which is thought to power the active galactic nucleus of this model:
"Mach’s principle is the only sane way to understand the origin of inertia in the universe and we must restrict to solutions of Einstein’s equations which are Machian. In particular the Kerr solution is not a valid model for the metric near a rotating mass and we need to find a Machian rotation metric (an MR–metric) to replace it. The fact that it is possible to construct non-Machian solutions to Einstein’s equations does not imply that Mach’s principle is somehow incompatible with Einstein. The Kerr metric is based on a particular set of boundary conditions, i.e. rotating central mass and flat, zero mass, at infinity. These boundary conditions are in themselves non-Machian and hence any solution which fits them must also break Mach’s principle. With good boundary conditions there may well be Machian solutions and this is what we hope to find."
In the context of the galactic model being reviewed, the description of what is called the galactic rotation curve appears to be of key importance. However, the idea of inertial rotation and frame dragging are often linked with the ideas encapsulated within Mach’s principles. By way of some initial historical background, Ernst Mach (1838-1916) was a physicist and philosopher, who is now often remember for his ideas about the possible root cause of inertial mass:
The inertial mass of a body is caused by its interactions with the other bodies in the universe.
While many of Mach’s ideas are said to have influenced Einstein in his formulation of general relativity, the full scope of Mach’s work in this area can often appear to be contradictory. However, the complications surrounding many of these ideas will not be address directly, although the following paper may provide some insight to many of the wider issues under debate:
The Lense–Thirring Effect and Mach’s Principle:
"By popular usage Mach’s principle has acquired a range of meanings, some of which are in conflict with each other. Mach’s writings have been a source of inspiration to many, including Einstein. We hope that our effort at distinguishing between existing versions of Mach’s Principle will serve to clarify ideas and eliminate needless controversy."
However, the physics of rotational frames might also be introduced in terms of the work of Georges Sagnac (1869-1928) and his discovery of what has become known as the ‘Sagnac Effect’ . In 1913, Sagnac developed an idea that would allow the measurement of rotation around some central point at some given radius [r], although the following description is somewhat updated, both in terms of technology and scope. Imagine an optical fibre is arranged in a circle, with radius [r], into which photons are then transmitted in both directions, such that the time taken for each direction can be measured. If they return simultaneously, the system is said to be non-rotating, otherwise the difference in the time taken is said to be a measure of how fast the device is rotating around some central point at radius [r]. The basic effect might be visualised as follows:
Note, for the purposes of this conceptual discussion, we will not worry about the refractive index [n] of the fibre affecting the speed of light [c]. The following derivation will now try to quantify the Sagnac effect in terms of the flat spacetime geometry defined by the Minkowski metric, where [r] now represents the coordinate orbital radius:
Given that the form of this equation can be referenced through the link provided above, the various parameters in  will not be detailed at this point. While this metric has only limited application when considering a galactic model driven by a rotating black hole surrounded by charged plasma, we might continue by transforming  into a pseudo-rotating metric by changing [dφ] to [dφ-ω.dt], where [ω] represents some angular velocity around the orbital axis.
We might immediately simplify  by recognising that for the purposes of this general discussion, we might set [ds=dr=dθ =0] for a photon in a circular equatorial orbits:
However, we want to solve this equation in terms of the overall rate of rotation, which can be done by dividing through by [dt2]:
At this point, it might be seen that  is now in the form of a quadratic equation [ax2+bx+c], which can be solved as follows:
As such, we might interpret the term [c/r] in  as the angular velocity of light in the case of a rotating system, although in terms of measurements, we would like to know the time taken in each direction:
Based on the form in , we might see that as the rotational velocity [ω] approaches zero, then the time difference also approaches zero, such that [∆t] can be seen to be a function of its angular velocity [ω]. While the discussion of the Sagnac effect might give us some basic insight to some of the mechanisms at work within rotating systems, it is clear that any galactic model will need to be extended beyond the flat spacetime of the Minkowski metric. However, we will defer this aspect to another discussion entitled ‘Black Hole Models’. For now, we might recognise that the work of Sagnac and Einstein both post-dated the work of Mach, such that we might appreciate that Mach was not really in a position to quantify his ideas beyond a certain amount of philosophical speculation. However, later, when Einstein published his general theory of relativity, in 1916, Mach’s principle started to be considered in terms of a spacetime geometry that was affected by the presence of matter. While the previous derivation linked to the Minkowski metric in  describes the idea of rotation in flat spacetime without any implied gravitational mass [M], we might realise that any rotational effects could still be inferred by reviewing the form of , when reduced to the equatorial plane, i.e. θ=90, where [dθ] and [dr] are both zero:
In the context of , the idea of rotation is linked to the coefficients [gφφ] and [gφt], where [gφt] is a new factor created by the introduction of the relative angular frame velocity [ω]. As such, the frame dragging angular velocity [ωD] might be defined as the ratio of [gφt] to [gφφ].
However, the form of  does not quantify the mass [M], which is ultimately the root cause of the frame dragging effect. To address this issue, we might be able to introduce the idea of a rotation mass [M] into  without changing its basic form as follows:
The final form in  is based on the fact that [MωR2] can be described as the angular momentum [J] of the rotating mass [M] at its radius [R], where [R] is assumed to be much larger than the Schwarzschild radius [Rs] for the moment. Given that the nature of  is not explicitly describing a black hole, where the physical radius [R] of mass [M] might be somewhat ambiguous, the form of  is said to describe weak-field frame dragging, which is often linked to the Lense-Thirring metric. However, the main purpose of these equations is simply to be able to quantify two initial examples of frame dragging in terms of the Earth and a galactic black hole with a mass of [~1012] solar masses, as assumed by the Rouke model under review. In , we see that the frame dragging velocity will be a function of the coordinate radius [r], while the angular momentum [J=MωR2] is define by the rotation of the Earth’s mass [ME] at its physical radius [RE]. The following graph shows how the angular [ωD] and tangential [vD] velocity change as a function of [r/RE].
In some respects, the plot of the tangential velocity [vd] is more intuitive than the angular velocity [ωd] and shows that even close to the Earth, the frame dragging velocity is incredibly small, such that its direct measurement is virtually impossible. However, as indicated, making a similar statement for a super-massive black hole ‘singularity’ hiding behind an event horizon at [Rs] might be subject to certain conceptual ambiguities. In the context of the galactic model being proposed, the black hole has an assumed mass [M] of ~1012 solar masses with an associated Schwarzschild radius [Rs] of 2.98*1015 metres.
But how do we quantify the rotation of a black hole?
Again, examination of  suggests that the rotation is quantified in terms of its angular momentum [J=MvR]. However, a black hole might be said to have no physical radius, only a centre of mass surrounded by an event horizon at [Rs] at which the escape velocity exceeds the speed of light [c].
So how can a black hole be described as rotating?
It might be inferred from the conservation of angular momentum [J] that if this quantity existed in a rotating star before its collapse into a black hole, it must be conserved after it collapses behind the event horizon. Equally, as the spinning star begins to collapse, i.e. its radius reduces, such that its rotational velocity [v] must increase in order to maintain its angular momentum [J=MvR]. In fact, calculations suggest that many black holes may ‘rotate’ with a velocity [v] approaching the speed of light [c]. In some respects, the description of this rotation is difficult to explain in the vocabulary of classical physics and possibly not much easier when using the mathematical semantics of general relativity. In a sense, the idea of ‘physical’ mass, as defined by kilograms, ceases to have any real meaning for an black hole and we are forced towards a description of an energy vortex of spacetime, although the idea of a Newtonian centre of mass remains. While a more detailed description will be attempted in the subsequent discussion of the Kerr metric, it might be worth outlining some of the issues at this stage, starting with the notion of geometric mass [Μ]:
In , we see that the geometric mass has units of metres, not kilograms, which we might also compare against the equation for the Schwarzschild radius:
As such,  first suggests that the physical mass [M=kg] of a black hole can be converted into the geometric mass [Μ=metres], which  then quantifies in terms of the distance radius [Rs/2]. While possibly not the preferred description of general relativity, it might be said that the rotational velocity [v] of the black hole of mass [M] is defined at a radius [Rs/2], such that we might now define the angular momentum of our galactic black hole example on the basis of the most extreme case, where [v=c]. Again, we might visualise this extreme example in terms of a graph:
While the shape of the curves does not look any different from the previous graph, it can be see that the tangential velocity, on the red-scale on the right, is now shown as a ratio [v/c]. As such, the spacetime surrounding such an extreme black hole could be dragged around at incredible velocities. However, what also needs to be highlighted, at this stage, is that even a Schwarzschild radius [Rs], as large as used in the example above, only extends out ~0.31 lightyears into a galaxy that has an estimated radius of some 50,000 lightyears. As such, the extreme relativistic frame dragging effects will only exist in the very heart of the Active Galactic Nucleus (AGN), although it is still assumed to be the power source generator for the galactic dynamics subsequently seen at larger radii. The inference of rotation inside the event horizon [Rs] has also not really been addressed, as this aspect is deferred to later discussions. While the Rouke model rejects the Kerr metric on the grounds that it cannot align to Mach’s principle, the analysis subsequently provided by this metric might still give a better insight to the scope of the issues linked to frame dragging around a super-massive black hole.